Function eregi() is deprecated [duplicate]

preg_match expects its regex argument to be within a pair delimiters.

So try:

if ( ! preg_match("#convert$#i", $this->library_path)) {
        if ( ! preg_match("#/$#i", $this->library_path)) 
                $this->library_path .= "/";

        $this->library_path .= 'convert';
}

if (preg_match("#gd2$#i", $protocol)) {                                         
        $protocol = 'image_process_gd'; 
}     

It seems you just forgot the Delimiter

preg_match("~/$~", $this->library_path)

and

preg_match("~gd2$~i", $protocol)

But in both cases you should consider not using regular expressions, because they are oversized here

$this->library_path[strlen($this->library_path) - 1] == '/'
substr($protocol, -3) == 'gd2'

If you are just checking for the presence of one string inside another, you should just use the strpos() function. eg:

if(strpos('convert', $this->library_path) !== false) {
    // code here
}

Update: Misread you want to check for it at the END of a string, this is still possible without Regex by using substr():

if(substr($this->library_path, -7) == 'convert'  {
    //code here
}

Where 7 is the length of convert, you could use a strlen and subtract it from 0 to get this number dynamically. This won't start any regex so is much more efficient.